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Effective Mathematics Interventions

Diane P. Bryant, Ph.D. Brian R. Bryant, Ph.D.

Kathleen Hughes Pfannenstiel, Ph.D. Meadows Center for Preventing Educational Risk:

Mathematics Institute

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Research to Practice: Literature

•  Skills in mathematics have been low as compared to other subject areas

•  Low enrollment in more advanced mathematics classes

•  Developmental delays –  Flexibility in using numbers

• Number sense • Mental number lines

–  Arithmetic combinations and equations of numbers –  Decomposing numbers

•  Lack conceptual, deep understanding of mathematics •  Lack procedural fluidity and strategy use

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National Assessment of Educational Progress (2009) •  Assessment in 5 areas:

–  Number and Operations –  Measurement –  Geometry –  Data Analysis –  Algebra

•  Scores in grade 4 have increased since 1990, but are not significantly different since 2007 –  82% performing at the basic level –  39% performing at the proficient level –  6% performing at the advanced level

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NMAP Recommendations

•  Focused mathematics curriculum to meet the critical foundational needs for algebraic readiness –  Fluency with whole number computations –  Proficiency with Fractions –  Aspects of Geometry and Measurement

•  Core instruction is not enough! –  Strategy instruction –  Explicit and systematic instruction –  Conceptual understanding

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Progression of Mathematics Skills Towards Algebra Readiness, from Grades 5–8

Using fractions and decimals to

represent, compare and order quantities, and solve problems

Connecting ratios, rates, and proportions to

multiplication and division and using

these concepts and operations to solve problems involving

proportional relationships

Representing and applying

proportionality

Representing, applying, and

analyzing proportionality

Mathematical Procedures and Skills

Real numbers, fractions, decimals, use of

algorithms

Proficiency with addition, subtraction,

multiplication, and division of whole number

algorithmsDeveloping proficient use of whole number

division standard algorithms to solve

problems

Developing an understanding of and fluency with

addition, subtraction, multiplication, and division of fractions

and decimals

Using rational numbers and

operations in a variety of contexts to

solve problems

Using proportionality and linear equations

to solve problems

Using formulas to find area, volume,

perimeter

Using estimation to examine solutions of arithmetic problems

Solving problems including

determining angles, areas, perimeters,

and volumes

Problem solving including similarity,

congruency, Pythagorean

Theorem, and complex volumes

Generalized ThinkingMultiplicative/

Proportional thinking

Generalized notation

Generalization of algorithms

Represent and compare whole numbers on the

number line

Understanding number lines, line

graphs, and number plots

Using expressions and equations to

represent situations involving rational

numbers

Using expressions and equations in a variety of

contexts, including measurement, conversions,

probability, and data analysis

Using expressions, equations, functions, and the real number system to represent and solve problems

in a variety of contexts

©2011 University of Texas System

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National Assessment of Educational Progress (NAEP), 2009

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

4th grade 8th grade

Students with Disabilities

Students without Disabilities

Percentages of Students who Scored “At or above Proficient” National Assessment of Educational Progress

Mathematics Performance in Algebra Readiness

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Mathematics Performance in Algebra Readiness •  TAKS, State Data: Mathematics: Grade 5

Total Tested: African-American:

Hispanic: White:

Eco-Dis: LEP:

Special Ed:

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Mathematics Performance in Algebra Readiness •  TAKS, State Data: Mathematics: Grade 10

Total Tested: African-American:

Hispanic: White:

Eco-Dis: LEP:

Special Ed:

Total Tested: African-American:

Hispanic: White:

Eco-Dis: LEP:

Special Ed:

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Procedures & Features: Tier II •  Identify an instructional sequence

–  Foundational –  Multiple opportunities to practice within the lessons

• Teach specific strategies • Build procedural knowledge

–  Increase student engagement •  Identify and teach prerequisite knowledge to build

–  Mastery –  Fluency

•  Quick pace –  Use of time to stay on-task –  Behavior management

•  Error correction and scaffolds

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Procedures & Features: Tier II •  Opportunities to make, show, write number concepts •  Enhance core curriculum through problem solving •  Regular, consistent intervention

–  4-5 days per week –  20-30 minutes

•  Progress monitoring –  Daily

•  Independent Practice • 1-2 minutes • Reflect material taught

–  Weekly/Bi-weekly • Aim Checks • Generalization

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Components of Explicit, Systematic Instruction

•  All aspects of instruction include –  Script –  Time –  Quick pace –  Mix of choral and independent responses –  Teacher talk decreases throughout lesson

•  Preview/Cumulative Review (2-3 minutes) –  Sets the tone for lesson –  Purpose of lesson –  Allows time to remediate/review skills –  Choral and individual responses

•  Modeled Practice (2-3 minutes) –  Teaches the skill explicitly –  Safe…students practice alongside or immediately following

teacher directions –  Quick pace –  Small steps

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Components of Explicit Instruction •  Guided Practice (6-8 minutes)

–  Similar to Modeled Practice –  Increase individual time for practice –  Provide error correction/scaffolds –  Transaction from concrete to pictorial and/or

pictorial to abstract –  Teaches how to complete the daily check

•  Independent Practice (2-3 minutes) –  Fluency –  Immediate feedback –  Error correction

•  Total Time: 12-17 minutes

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Multiple Representations •  Concrete: Modeling/Guided Practice

–  Cubes –  Counters –  Base-ten/Place value materials (units, rods, flats) –  Dot cubes

•  Pictorial: Guided Practice/Independent Practice –  Five frames –  Ten frames –  Hundreds charts –  Number lines

•  Abstract: Guided Practice/Independent Practice –  Numbers

•  Mats (can be used across all representations) –  Part-part-whole –  Fact family –  Strategy mats

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Student Verbalizations

•  Student Verbalizations includes students’ thinking aloud about their problem solving approaches, mathematical understanding, or promoting mathematical discourse (Gersten et al., 2009) – Questions – Neighbor-share – Choral response

• Wipe boards • Multiple choice answers

–  Identifying mistakes

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Student Verbalizations

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Visual Representations

•  Using Visual Representations used during instruction include those used by the teacher to model problem solving as well as student use of manipulatives (Gersten et al., 2009).

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Visual Representations

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3 Tier Mathematics Model

•  Free to Texas Educators at http://3tiermathmodel.org/

•  Username: Texas Teacher •  Password: mathematics

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Content and Skills: Grades K-2nd •  Word Problem Solving:

–  Strategy to solve all types –  Different types of problems –  Extraneous information –  Multiple steps, contextualized

•  Number Knowledge and Relationships –  Counting: Rote, Rational, Counting Up/Back, Skip (2, 5, 10) –  Number Recognition & Writing: 0-20 (kinder) 0 – 99 (1st); 0 – 999 (2nd

grade) •  Number Relationships of greater than/less than/equal to

–  Relationships of one and two more than/less than –  Anchoring Numbers to 5 & 10 frames –  Part-part-whole Relationships (e.g., ways to represent numbers)

•  Numeric Sequencing –  Number line, mental number line –  Math flexibility –  Ordering numbers

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Grades K-2nd (cont.) •  Base 10 & Place Value

–  Making and counting: •  Groups of tens and ones (1st grade) •  Groups of hundreds, tens, and ones (2nd grade) •  Using base-ten language (3 hundreds, 0 tens, 6 ones) and

standard language (306) to describe place value –  Reading and writing numbers to represent base ten models –  Naming the place value held by digits in numbers

•  Addition & Subtraction Combinations –  Identity Element and Properties –  Fact Families –  Counting & Decomposition Strategies

•  Addition: count on, [+ 0, + 1, + 2], doubles, doubles +1, make 10 + more

•  Subtraction: count down [-0, -1, -2, -3], count on

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Intervention Modules

Grade 3 •  Place Value Concepts •  Addition & Subtraction of

Whole Numbers •  Multiplication & Division

Concepts •  Fraction Concepts

Grade 4 •  Multiplication & Division

Strategies •  Multiplication & Division

of Whole Numbers •  Modeling, Comparing, &

Ordering Fractions •  Fraction & Decimal

Relationships

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• The students is expected to: • Use place value to read, write (in symbols and words) and describe the value of whole numbers through 999,999. (3.1 A, supporting) • Use place value to compare and order whole numbers through 9,999. (3.1 B, supporting) • Identify and extend whole-number patterns. (3.6 A; supporting) • Use data to describe events as more likely than, less likely than, or equally likely as. (3.13 C, supporting) • Applies Grade 3 math to solve problems connected to everyday experiences in and outside of school. The student is expected to

understand the problem, make a plan, carry out the plan and evaluate the solution for reasonableness. (3.14 A; B; C; D) • Can relate informal language to mathematical language and symbols. (3.15 B)

Module A: Place Value Concepts

• The students is expected to: • Model addition and subtraction using pictures, words and numbers (3.3 A, supporting) • Select addition or subtraction and use the operation to solve problems involving whole numbers through 999 (3.3 B, readiness) • Round whole numbers to the nearest ten or hundred to approximate reasonable results in problem situations. (3.5 A, supporting) • Use strategies including rounding and compatible numbers to estimate solutions to addition and subtraction problems. (3.5 B,

supporting) • Identify and extend whole-number patterns to make predictions and solve problems (3.6 A, supporting) • Applies Grade 3 math to solve problems connected to everyday experiences in and outside of school. The student is expected to

understand the problem, make a plan, carry out the plan and evaluate the solution for reasonableness. (3.14 A; B; C; D) • Can relate informal language to mathematical language and symbols. (3.15 B)

Module B: Addition & Subtraction

of Whole Numbers

• The students is expected to: • Learn and apply multiplication facts through 12 by 12 using concrete models and objects (3.4 A, supporting) • Solve and record multiplication problems. (3.4 B, readiness) • Use models to solve division problems and use number sentences to record the solutions. (3.4 C, supporting) • Identify patterns in multiplication facts using concrete objects, pictorial models or technology. (3.6 B, supporting) • Identify patterns in related multiplication and division sentences (fact families). (3.6 C, supporting) • Applies Grade 3 math to solve problems connected to everyday experiences in and outside of school. The student is expected to

understand the problem, make a plan, carry out the plan and evaluate the solution for reasonableness. (3.14 A; B; C; D) • Can relate informal language to mathematical language and symbols. (3.15 B)

Module C: Multiplication and Division Concepts

• The students is expected to: • Compare fractional parts of whole objects or sets of objects in a problem situation using concrete models (3.2 B, readiness) • Use fraction names and symbols to describe fractional parts of whole objects or sets of objects (3.2 C, readiness) • Construct concrete models of equivalent fractions for fractional parts of whole objects. (3.2 D) •  Student recognizes that a line can be used to represent numbers and fractions and their properties and relationships. Can locate

and name points on number line using whole numbers and fractions, including halves and fourths. (3.10 A, readiness) • Applies Grade 3 math to solve problems connected to everyday experiences in and outside of school. The student is expected to

understand the problem, make a plan, carry out the plan and evaluate the solution for reasonableness. (3.14 A; B; C; D) • Can relate informal language to mathematical language and symbols. (3.15 B)

Module D: Fraction Concepts

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•  The students is expected to: •  Use multiplication and division to solve problems without technology. (4.4 D, E, readiness) •  Model factors and products using arrays and area models. (4.4 A, supporting; 4.4 B, supporting) •  Represent multiplication and division situations in picture, word and number form. (4.4 B, supporting) •  Use multiplication and division to solve problems (4.4 D, E, readiness) •  Use patterns and relationships to develop strategies to remember basic multiplication and division facts (4.6 A, supporting ) •  Represent multiplication and division situations in picture, word, and number form. (4.4 B, supporting) •  Recall and apply multiplication facts through 12 x 12. (4.4 C, supporting) •  Applies Grade 4 math to solve problems connected to everyday experiences in and outside of school. The student is expected to understand the problem, make a plan, carry out the plan

and evaluate the solution for reasonableness. (4.14 A; B; C; D) •  Can relate informal language to mathematical language and symbols. (4.15 B) •  Make generalizations from patterns or sets of examples and nonexamples (4.16 A) •  Justify why an answer is reasonable and explain the solution process. (4.16 B)

Module A: Multiplication &

Division Strategies

•  The students is expected to: •  Use multiplication and division to solve problems (no more than 2 digits times 2 digits or one digit divisor and three-digit dividends without technology). (4.4 D; E, readiness) •  Model factors and products using arrays and area models (4.4 A, supporting) •  Represent multiplication and division situations in picture, word and number form. (4.4 B, supporting) •  Use strategies including rounding and compatible numbers to estimate solutions to multiplication and division problems (4.5 B, supporting) •  Use patterns to multiply by 10 and 100. (4.6 B, supporting) •  Applies Grade 4 math to solve problems connected to everyday experiences in and outside of school. The student is expected to understand the problem, make a plan, carry out the plan

and evaluate the solution for reasonableness. (4.14 A; B; C; D) •  Can relate informal language to mathematical language and symbols. (4.15 B) •  Make generalizations from patterns or sets of examples and nonexamples (4.16 A) •  Justify why an answer is reasonable and explain the solution process. (4.16 B)

Module B: Multiplication & Division of Whole

Numbers

•  The students is expected to: •  Model fraction quantities greater than one using concrete objects and pictorial models. (4.2 B, supporting) •  Use concrete objects and pictorial models to generate equivalent fractions. (4.2 A, supporting) •  To locate and name points on a number line using whole numbers and fractions such as halves and fourths. (4.10 A, readiness) •  Compare and order fractions using concrete objects and pictorial models. (4.2 C, supporting) •  Applies Grade 4 math to solve problems connected to everyday experiences in and outside of school. The student is expected to understand the problem, make a plan, carry out the plan

and evaluate the solution for reasonableness. (4.14 A; B; C; D) •  Can relate informal language to mathematical language and symbols. (4.15 B) •  Make generalizations from patterns or sets of examples and nonexamples (4.16 A) •  Justify why an answer is reasonable and explain the solution process. (4.16 B)

Module C: Modeling, Comparing &

Ordering Fractions

•  The students is expected to: •  Relate decimals to fractions that name tenths and hundredths using concrete objects and pictorial models.(4.2 D, readiness) •  Locate and name points on a number line using whole numbers, fractions such as halves and fourths, and decimals such as tenths. (4.10 A, readiness) •  Relate decimals to fractions that name tenths and hundredths using concrete objects and pictorial models. (4.2 D, readiness) •  Use place value to read, write, compare and order decimals involving tens and hundredths, including money, using concrete objects and pictorial models. (4.1 B, readiness; 4.2 C,

supporting) •  Applies Grade 4 math to solve problems connected to everyday experiences in and outside of school. The student is expected to understand the problem, make a plan, carry out the plan

and evaluate the solution for reasonableness. (4.14 A; B; C; D) •  Can relate informal language to mathematical language and symbols. (4.15 B) •  Make generalizations from patterns or sets of examples and nonexamples (4.16 A) •  Justify why an answer is reasonable and explain the solution process. (4.16 B)

Module D: Fraction & Decimal

Relationships

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Intervention 6th – 9th Grade

•  Project Share –  MSTAR: Middle School Intervention

• Facts & Patterns in Multiplication and Division • Proportionality • Ratios and Rates • Equivalent Fractions

–  H.S. Algebra Intervention Modules • Variables • Expressions, Equations, & Equivalence

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Students Who Struggle…. NMAP Progressions

~ Taken from National Mathematics Advisory Panel, 2008, p. 16, 20